HINT: <no title>
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There are two ways of proving that triangles are similar:
- Prove that they are equiangular.
- Prove that their sides are in proportion.
Which of these strategies is more appropriate for this question?
STEP: Match up sides and find the proportionality constant
[−3 points ⇒ 1 / 4 points left]
There are two ways of proving that triangles are similar:
- Prove that they are equiangular.
- Prove that their sides are in proportion.
In this question, we have more information about the
sides of the triangles than about the angles. So, we will prove that all
of the sides are in proportion.
We must show that each side must be multiplied by the same number to make its matching side in the other triangle.
Firstly, we must match up the sides:
- The longest sides (NM and JK) match up.
- The shortest sides (NP and JL) match up.
- PM and KL must match up because they are the only sides left.
Now we will work out what each side in ΔJKL should be multiplied by:
NMJKMPKLNPJL=3514=2,5=27,511=2,5=208=2,5
This means that any side in ΔJKL must be multiplied by 2,5 to make its matching side in ΔNMP.
For example, 14×2,5=35.
We call this number the proportionality constant.
If any one of these division sums had
given us a different answer, the sides of the triangles would not have
been in proportion. Then, the triangles would not be similar.
If your calculations tell you that the triangles are not similar in a question where you are asked to prove similarity, you probably made a mistake somewhere. Check your working out!
STEP: Prove similarity
[−1 point ⇒ 0 / 4 points left]
Now that we have matched up the sides, and worked out the
proportionality constant, we just need to put the information into a
formal similarity proof:
In ΔJKL and ΔNMP:
- NM÷JK = 3514 = 2,5
- MP÷KL = 27,511 = 2,5
- NP÷JL = 208 = 2,5
∴ΔJKL|||ΔNMP (sides of Δ in prop)
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